Background
knowledge
Contents: A Surds and radicals
B Scientific notation (standard form)
C Number systems and set notation
D Algebraic simplification
E Linear equations and inequalities
F Modulus or absolute value
G Product expansion
H Factorisation
I Formula rearrangement
J Adding and subtracting algebraic fractions
K Congruence and similarity
L Pythagoras’ theorem
M Coordinate geometry
N Right angled triangle trigonometry
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
25
75
95
25
75
95
25
75
95
25
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5
5
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\001IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:01 AM BEN
, 2 BACKGROUND KNOWLEDGE
This chapter contains material that is assumed knowledge for the course. It does not cover
all assumed knowledge, as other necessary work is revised within the chapters.
A SURDS AND RADICALS
A radical is any number which is written with the radical sign p .
p p p p
A surd is a real, irrational radical such as 2, 3, 5 or 6. Surds are present in solutions
p
to some quadratic equations. 4 is a radical but is not a surd as it simplifies to 2.
p p p
a is the non-negative number such that a £ a = a.
p p
Properties: ² a is never negative, so a > 0.
p
² a is meaningful only for a > 0.
r
p p p
² ab = a £ b for a > 0 and b > 0.
p
a a
² = p for a > 0 and b > 0.
b b
Example 1 Self Tutor
p p p
Write as a single surd: a 2£ 3 b p18
6
q
p p p p
a 2£ 3 b p18 or p18
p 6 6
p p
= 2£3 6£
p 3
p = 18 = 6
6
= 6 p p
= 3 = 3
EXERCISE A
1 Write as a single surd or rational number:
p p p p p p p
a 3£ 5 b ( 3)2 c 2 2£ 2 d 3 2£2 2
p p p p p
e 3 7£2 7 f p12 2
g p12
6
h p18
3
Compare with
Example 2 Self Tutor 2x ¡ 5x = ¡3x
p p p p
Simplify: a 3 3+5 3 b 2 2¡5 2
p p p p
a 3 3+5 3 b 2 2¡5 2
p p
= (3 + 5) 3 = (2 ¡ 5) 2
p p
=8 3 = ¡3 2
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
25
75
95
25
75
95
25
75
95
25
75
95
5
5
5
5
cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\002IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:07 AM BEN
, BACKGROUND KNOWLEDGE 3
2 Simplify the following mentally:
p p p p p p p p
a 2 2+3 2 b 2 2¡3 2 c 5 5¡3 5 d 5 5+3 5
p p p p p p p p p
e 3 5¡5 5 f 7 3+2 3 g 9 6 ¡ 12 6 h 2+ 2+ 2
Example 3 Self Tutor
p p
Write 18 in the form a b where a and b are integers and a is as large as possible.
p
18
p
= 9£2 f9 is the largest perfect square factor of 18g
p p
= 9£ 2
p
=3 2
p
3 Write the following in the form a b where a and b are integers and a is as large as
possible:
p p p p
a 8 b 12 c 20 d 32
p p p p
e 27 f 45 g 48 h 54
p p p p
i 50 j 80 k 96 l 108
Example 4 Self Tutor
p p
Simplify: 2 75 ¡ 5 27
p p
2 75 ¡ 5 27
p p
= 2 25 £ 3 ¡ 5 9 £ 3
p p
= 2£5£ 3¡5£3£ 3
p p
= 10 3 ¡ 15 3
p
= ¡5 3
4 Simplify:
p p p p p p
a 4 3 ¡ 12 b 3 2 + 50 c 3 6 + 24
p p p p p p p
d 2 27 + 2 12 e 75 ¡ 12 f 2 + 8 ¡ 32
Example 5 Self Tutor
Write p9 without a radical in the p9
3 3 p
denominator. = p £ p3
9
p3 3
9 3
=
p
3
=3 3
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
25
75
95
25
75
95
25
75
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25
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\003IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:12 AM BEN
, 4 BACKGROUND KNOWLEDGE
5 Write without a radical in the denominator:
a p12 b p63 c p7
2
d 10
p
5
10 18 12 p5
e p
2
f p
6
g p
3
h 7
p
14 2p 3
i p
7
j 2
B SCIENTIFIC NOTATION
(STANDARD FORM)
Scientific notation (or standard form) involves writing any given number as
a number between 1 and 10, multiplied by a power of 10,
i.e., a £ 10k where 1 6 a < 10 and k 2 Z .
Example 6 Self Tutor
Write in standard form: a 37 600 b 0:000 86
a 37 600 = 3:76 £ 10 000 fshift decimal point 4 places to the
= 3:76 £ 104 left and £ 10 000g
b 0:000 86 = 8:6 ¥ 104 fshift decimal point 4 places to the
= 8:6 £ 10¡4 right and ¥ 10 000g
EXERCISE B
1 Express the following in scientific notation:
a 259 b 259 000 c 2:59 d 0:259
e 0:000 259 f 40:7 g 4070 h 0:0407
i 407 000 j 407 000 000 k 0:000 040 7
2 Express the following in scientific notation:
a The distance from the Earth to the Sun is 149 500 000 000 m.
b Bacteria are single cell organisms, some of which have a diameter of 0:0003 mm.
c A speck of dust has width smaller than 0:001 mm.
d The core temperature of the Sun is 15 million degrees Celsius.
e A single red blood cell lives for about four months. During this
time it will circulate around the body 300 000 times.
IB_SL-2ed
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100
100
100
50
50
50
50
0
0
0
0
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25
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75
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25
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\004IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:18 AM BEN
knowledge
Contents: A Surds and radicals
B Scientific notation (standard form)
C Number systems and set notation
D Algebraic simplification
E Linear equations and inequalities
F Modulus or absolute value
G Product expansion
H Factorisation
I Formula rearrangement
J Adding and subtracting algebraic fractions
K Congruence and similarity
L Pythagoras’ theorem
M Coordinate geometry
N Right angled triangle trigonometry
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
25
75
95
25
75
95
25
75
95
25
75
95
5
5
5
5
cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\001IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:01 AM BEN
, 2 BACKGROUND KNOWLEDGE
This chapter contains material that is assumed knowledge for the course. It does not cover
all assumed knowledge, as other necessary work is revised within the chapters.
A SURDS AND RADICALS
A radical is any number which is written with the radical sign p .
p p p p
A surd is a real, irrational radical such as 2, 3, 5 or 6. Surds are present in solutions
p
to some quadratic equations. 4 is a radical but is not a surd as it simplifies to 2.
p p p
a is the non-negative number such that a £ a = a.
p p
Properties: ² a is never negative, so a > 0.
p
² a is meaningful only for a > 0.
r
p p p
² ab = a £ b for a > 0 and b > 0.
p
a a
² = p for a > 0 and b > 0.
b b
Example 1 Self Tutor
p p p
Write as a single surd: a 2£ 3 b p18
6
q
p p p p
a 2£ 3 b p18 or p18
p 6 6
p p
= 2£3 6£
p 3
p = 18 = 6
6
= 6 p p
= 3 = 3
EXERCISE A
1 Write as a single surd or rational number:
p p p p p p p
a 3£ 5 b ( 3)2 c 2 2£ 2 d 3 2£2 2
p p p p p
e 3 7£2 7 f p12 2
g p12
6
h p18
3
Compare with
Example 2 Self Tutor 2x ¡ 5x = ¡3x
p p p p
Simplify: a 3 3+5 3 b 2 2¡5 2
p p p p
a 3 3+5 3 b 2 2¡5 2
p p
= (3 + 5) 3 = (2 ¡ 5) 2
p p
=8 3 = ¡3 2
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
25
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25
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25
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\002IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:07 AM BEN
, BACKGROUND KNOWLEDGE 3
2 Simplify the following mentally:
p p p p p p p p
a 2 2+3 2 b 2 2¡3 2 c 5 5¡3 5 d 5 5+3 5
p p p p p p p p p
e 3 5¡5 5 f 7 3+2 3 g 9 6 ¡ 12 6 h 2+ 2+ 2
Example 3 Self Tutor
p p
Write 18 in the form a b where a and b are integers and a is as large as possible.
p
18
p
= 9£2 f9 is the largest perfect square factor of 18g
p p
= 9£ 2
p
=3 2
p
3 Write the following in the form a b where a and b are integers and a is as large as
possible:
p p p p
a 8 b 12 c 20 d 32
p p p p
e 27 f 45 g 48 h 54
p p p p
i 50 j 80 k 96 l 108
Example 4 Self Tutor
p p
Simplify: 2 75 ¡ 5 27
p p
2 75 ¡ 5 27
p p
= 2 25 £ 3 ¡ 5 9 £ 3
p p
= 2£5£ 3¡5£3£ 3
p p
= 10 3 ¡ 15 3
p
= ¡5 3
4 Simplify:
p p p p p p
a 4 3 ¡ 12 b 3 2 + 50 c 3 6 + 24
p p p p p p p
d 2 27 + 2 12 e 75 ¡ 12 f 2 + 8 ¡ 32
Example 5 Self Tutor
Write p9 without a radical in the p9
3 3 p
denominator. = p £ p3
9
p3 3
9 3
=
p
3
=3 3
IB_SL-2ed
100
100
100
100
50
50
50
50
0
0
0
0
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\003IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:12 AM BEN
, 4 BACKGROUND KNOWLEDGE
5 Write without a radical in the denominator:
a p12 b p63 c p7
2
d 10
p
5
10 18 12 p5
e p
2
f p
6
g p
3
h 7
p
14 2p 3
i p
7
j 2
B SCIENTIFIC NOTATION
(STANDARD FORM)
Scientific notation (or standard form) involves writing any given number as
a number between 1 and 10, multiplied by a power of 10,
i.e., a £ 10k where 1 6 a < 10 and k 2 Z .
Example 6 Self Tutor
Write in standard form: a 37 600 b 0:000 86
a 37 600 = 3:76 £ 10 000 fshift decimal point 4 places to the
= 3:76 £ 104 left and £ 10 000g
b 0:000 86 = 8:6 ¥ 104 fshift decimal point 4 places to the
= 8:6 £ 10¡4 right and ¥ 10 000g
EXERCISE B
1 Express the following in scientific notation:
a 259 b 259 000 c 2:59 d 0:259
e 0:000 259 f 40:7 g 4070 h 0:0407
i 407 000 j 407 000 000 k 0:000 040 7
2 Express the following in scientific notation:
a The distance from the Earth to the Sun is 149 500 000 000 m.
b Bacteria are single cell organisms, some of which have a diameter of 0:0003 mm.
c A speck of dust has width smaller than 0:001 mm.
d The core temperature of the Sun is 15 million degrees Celsius.
e A single red blood cell lives for about four months. During this
time it will circulate around the body 300 000 times.
IB_SL-2ed
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100
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cyan magenta yellow black
:\HAESE\IB_SL-3ed\x-worksheets\BckgKnowl\004IB_SL-3_bg.CDR Wednesday, 21 March 2012 11:18:18 AM BEN
